Coherent structures theory for the generalized Kuramoto-Sivashinsky equation
Coherent structures theory for the generalized Kuramoto-Sivashinsky equation
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DOI：
10.1088/1742-6596/216/1/012018
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发表时间：
2010-03
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影响因子：
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通讯作者：
D. Tseluiko;S. Saprykin;S. Kalliadasis
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文献类型：
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作者：
D. Tseluiko;S. Saprykin;S. Kalliadasis
本文研究了有源色散耗散非线性介质中相干结构、相互作用和束缚态的形成。这种介质的原型是一个简单的弱非线性模型，广义Kuramoto-Sivashinsky（gKS）方程，它保留了任何涉及波演化的非线性过程的基本机制，即主要的非线性，不稳定性，稳定性和色散。我们发展了一个弱相互作用理论的孤立脉冲的gKS方程的脉冲和重叠函数的叠加表示的解决方案。我们推导出一个线性方程的重叠函数在附近的每个脉冲和项目的动态这个函数的线性化的相互作用算子的频谱的离散部分。这导致了一个耦合系统的常微分方程描述的脉冲的位置的演变。通过分析这个系统，我们证明了存在可数无穷或有限个束缚态的一个判据，这取决于方程中色散项的强度。理论研究结果证实了计算的完整方程。
We examine coherent structures interaction and formation of bound states in active–dispersive–dissipative nonlinear media. A prototype for such media is a simple weakly nonlinear model, the generalized Kuramoto-Sivashinsky (gKS) equation, that retains the fundamental mechanisms of any nonlinear process involving wave evolution, namely, a dominant nonlinearity, instability, stability and dispersion. We develop a weak interaction theory for the solitary pulses of the gKS equation by representing the solution as a superposition of the pulses and an overlap function. We derive a linearized equation for the overlap function in the vicinity of each pulse and project the dynamics of this function onto the discrete part of the spectrum of the linearized interaction operator. This leads to a coupled system of ordinary differential equations describing the evolution of the locations of the pulses. By analyzing this system, we prove a criterion for the existence of a countable infinite or finite number of bound states, depending on the strength of the dispersive term in the equation. The theoretical findings are corroborated by computations of the full equation.